9000034708
Level:
A
Consider the equation
\[
2x^{2} + 5px + 2 = 0
\]
with the real parameter \(p\).
Solve the equation for \(p = -\frac{4}
{5}\).
\(\left \{1\right \}\)
\(\left \{-1\right \}\)
\(\left \{0\right \}\)
\(\emptyset \)
Equations and Inequalities with Parameters
9000034710
Level:
A
Solve the following equation with a real parameter
\(t\), assuming
\(t\neq - 1\) and
\(t\neq 1\).
\[
x(t^{2} - 1) = t - 1
\]
\(\left \{ \frac{1}
{t+1}\right \}\)
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{0\right \}\)
9000034701
Level:
B
Identify a set of the values of the real parameter
\(m\) which
ensure that the equation
\[
\frac{m}
{x} - 8 = \frac{1}
{x} -\frac{m + 3}
{2}
\]
has solution \(x = 2\).
\(\left \{7\right \}\)
\(\left \{10\right \}\)
\(\left \{6\right \}\)
\(\left \{\frac{5}
{2}\right \}\)
9000034709
Level:
A
Consider equation
\[
p(2 - p)x = 4p
\]
with a real parameter \(p\).
Solve the equation for \(p = 2\).
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{ \frac{4}
{2-p}\right \}\)
\(\mathbb{R}\setminus \left \{0\right \}\)
9000034702
Level:
B
Identify the set of the values of the real parameter \(d\) for which the following equation has no solution in \(\mathbb{R}\).
\[
x^{2} - 2dx + 2d^{2} - 9 = 0
\]
\((-\infty ,-3)\cup (3,\infty )\)
\((-3,3)\)
\((3,\infty )\)
\((-\infty ,-3)\)
9000034703
Level:
B
Identify the set of the values of the real parameter \(t\) for which the following equation has two different real roots.
\[
x^{2} + (t + 2)x + 1 = 0
\]
\((-\infty ,-4)\cup (0,\infty )\)
\((-\infty ,-4)\)
\((-4,0)\)
\((0,\infty )\)
9000033704
Level:
B
Find the values of real parameter \(p\)
which ensure that the following quadratic equation has solutions with nonzero
imaginary part.
\[
px^{2} + 4x - p + 5 = 0
\]
\(p\in \left (1,4\right )\)
\(p\in [ 1,4] \)
\(p\in \left (-\infty ,1\right )\cup \left (4,\infty \right )\)
\(p\in \left (-\infty ,1\right ] \cup \left [ 4,\infty \right )\)
9000021706
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is bigger than \(10\).
\[
3x - 18 = \frac{10x - 4k}
{2}
\]
\(k\in (19,\infty )\)
\(k\in \{9\}\)
\(k\in (-\infty ,1)\)
\(k\in (9,\infty )\)
9000021707
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is positive.
\[
2kx + k = 4x + 3
\]
\(k\in (2,3)\)
\(k > 0\)
\(k\in (3,\infty )\)
\(k\in (-\infty ,3)\)